Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To find the inverse of the square root parent function [tex]\( F(x) = \sqrt{x} \)[/tex], we need to determine which function, when composed with the square root function, results in the identity function [tex]\( x \)[/tex].
The definition of an inverse function [tex]\( g(x) \)[/tex] of [tex]\( f(x) \)[/tex] is such that if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( g \)[/tex] and [tex]\( f \)[/tex] respectively, then [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Here, [tex]\( F(x) = \sqrt{x} \)[/tex]. Let's consider the given options to see which function is its inverse.
### Option A: [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
- First, let's verify if this function is the inverse:
1. Given [tex]\( y = \sqrt{x} \)[/tex], squaring both sides yields [tex]\( y^2 = x \)[/tex]. So, [tex]\( \sqrt{x} \)[/tex] undoes [tex]\( x^2 \)[/tex] when [tex]\( x \geq 0 \)[/tex].
2. Similarly, if [tex]\( F(x) = x^2 \)[/tex] where [tex]\( x \geq 0 \)[/tex], then [tex]\( F^{-1}(x) = \sqrt{x} \)[/tex].
Thus, for [tex]\( y = \sqrt{x} \)[/tex], if [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x^2 \geq 0 \)[/tex].
- [tex]\( F^{-1}(F(x)) = x \)[/tex] and [tex]\( F(F^{-1}(x)) = x \)[/tex].
So, [tex]\( F(x) = x^2 \)[/tex], [tex]\( x \geq 0 \)[/tex] correctly defines [tex]\( \sqrt{x} \)[/tex] as its inverse.
### Option B: [tex]\( F(x) = |x| \)[/tex]
- This function maps both positive and negative [tex]\( x \)[/tex] to their absolute values. However, the inverse of [tex]\( \sqrt{x} \)[/tex] should map back to a specific single value.
- The square root function is not the inverse of the absolute value function, because [tex]\( \sqrt{x} \)[/tex] does not undo [tex]\( |x| \)[/tex].
### Option C: [tex]\( F(x) = \frac{1}{\sqrt{x}} \)[/tex]
- This option represents a reciprocal transformation, which is not related to squaring or unsquaring a number.
- The square root function is not the inverse of the reciprocal of the square root.
### Option D: [tex]\( F(x) = x^2 \)[/tex]
- Without specifying the domain restriction [tex]\( x \geq 0 \)[/tex], the function [tex]\( F(x) = x^2 \)[/tex] includes negative values.
- For negative values, [tex]\( \sqrt{x} \)[/tex] is not defined, so [tex]\( F(x) = x^2 \)[/tex] without the domain restriction cannot be correct.
### Conclusion
Thus, the correct function that [tex]\( F(x) = \sqrt{x} \)[/tex] is the inverse of is:
A. [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
The definition of an inverse function [tex]\( g(x) \)[/tex] of [tex]\( f(x) \)[/tex] is such that if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( g \)[/tex] and [tex]\( f \)[/tex] respectively, then [tex]\( g(x) \)[/tex] is the inverse of [tex]\( f(x) \)[/tex].
Here, [tex]\( F(x) = \sqrt{x} \)[/tex]. Let's consider the given options to see which function is its inverse.
### Option A: [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
- First, let's verify if this function is the inverse:
1. Given [tex]\( y = \sqrt{x} \)[/tex], squaring both sides yields [tex]\( y^2 = x \)[/tex]. So, [tex]\( \sqrt{x} \)[/tex] undoes [tex]\( x^2 \)[/tex] when [tex]\( x \geq 0 \)[/tex].
2. Similarly, if [tex]\( F(x) = x^2 \)[/tex] where [tex]\( x \geq 0 \)[/tex], then [tex]\( F^{-1}(x) = \sqrt{x} \)[/tex].
Thus, for [tex]\( y = \sqrt{x} \)[/tex], if [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x^2 \geq 0 \)[/tex].
- [tex]\( F^{-1}(F(x)) = x \)[/tex] and [tex]\( F(F^{-1}(x)) = x \)[/tex].
So, [tex]\( F(x) = x^2 \)[/tex], [tex]\( x \geq 0 \)[/tex] correctly defines [tex]\( \sqrt{x} \)[/tex] as its inverse.
### Option B: [tex]\( F(x) = |x| \)[/tex]
- This function maps both positive and negative [tex]\( x \)[/tex] to their absolute values. However, the inverse of [tex]\( \sqrt{x} \)[/tex] should map back to a specific single value.
- The square root function is not the inverse of the absolute value function, because [tex]\( \sqrt{x} \)[/tex] does not undo [tex]\( |x| \)[/tex].
### Option C: [tex]\( F(x) = \frac{1}{\sqrt{x}} \)[/tex]
- This option represents a reciprocal transformation, which is not related to squaring or unsquaring a number.
- The square root function is not the inverse of the reciprocal of the square root.
### Option D: [tex]\( F(x) = x^2 \)[/tex]
- Without specifying the domain restriction [tex]\( x \geq 0 \)[/tex], the function [tex]\( F(x) = x^2 \)[/tex] includes negative values.
- For negative values, [tex]\( \sqrt{x} \)[/tex] is not defined, so [tex]\( F(x) = x^2 \)[/tex] without the domain restriction cannot be correct.
### Conclusion
Thus, the correct function that [tex]\( F(x) = \sqrt{x} \)[/tex] is the inverse of is:
A. [tex]\( F(x) = x^2 \)[/tex], where [tex]\( x \geq 0 \)[/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
